Theorem rngolz 33721

Description: The zero of a unital ring is a left-absorbing element. (Contributed by FL, 31-Aug-2009.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ringlz.1	|- Z = (GId ` G )
ringlz.2	|- X = ran G
ringlz.3	|- G = ( 1st ` R )
ringlz.4	|- H = ( 2nd ` R )

Assertion

Ref	Expression
rngolz	|- ( ( R e. RingOps /\ A e. X ) -> ( Z H A ) = Z )

Proof of Theorem rngolz

Step	Hyp	Ref	Expression
1		ringlz.3	. . . . . . 7 |- G = ( 1st ` R )
2	1	rngogrpo 33709	. . . . . 6 |- ( R e. RingOps -> G e. GrpOp )
3		ringlz.2	. . . . . . . 8 |- X = ran G
4		ringlz.1	. . . . . . . 8 |- Z = (GId ` G )
5	3, 4	grpoidcl 27368	. . . . . . 7 |- ( G e. GrpOp -> Z e. X )
6	3, 4	grpolid 27370	. . . . . . 7 |- ( ( G e. GrpOp /\ Z e. X ) -> ( Z G Z ) = Z )
7	5, 6	mpdan 702	. . . . . 6 |- ( G e. GrpOp -> ( Z G Z ) = Z )
8	2, 7	syl 17	. . . . 5 |- ( R e. RingOps -> ( Z G Z ) = Z )
9	8	adantr 481	. . . 4 |- ( ( R e. RingOps /\ A e. X ) -> ( Z G Z ) = Z )
10	9	oveq1d 6665	. . 3 |- ( ( R e. RingOps /\ A e. X ) -> ( ( Z G Z ) H A ) = ( Z H A ) )
11	1, 3, 4	rngo0cl 33718	. . . . . 6 |- ( R e. RingOps -> Z e. X )
12	11	adantr 481	. . . . 5 |- ( ( R e. RingOps /\ A e. X ) -> Z e. X )
13		simpr 477	. . . . 5 |- ( ( R e. RingOps /\ A e. X ) -> A e. X )
14	12, 12, 13	3jca 1242	. . . 4 |- ( ( R e. RingOps /\ A e. X ) -> ( Z e. X /\ Z e. X /\ A e. X ) )
15		ringlz.4	. . . . 5 |- H = ( 2nd ` R )
16	1, 15, 3	rngodir 33704	. . . 4 |- ( ( R e. RingOps /\ ( Z e. X /\ Z e. X /\ A e. X ) ) -> ( ( Z G Z ) H A ) = ( ( Z H A ) G ( Z H A ) ) )
17	14, 16	syldan 487	. . 3 |- ( ( R e. RingOps /\ A e. X ) -> ( ( Z G Z ) H A ) = ( ( Z H A ) G ( Z H A ) ) )
18	2	adantr 481	. . . 4 |- ( ( R e. RingOps /\ A e. X ) -> G e. GrpOp )
19		simpl 473	. . . . 5 |- ( ( R e. RingOps /\ A e. X ) -> R e. RingOps )
20	1, 15, 3	rngocl 33700	. . . . 5 |- ( ( R e. RingOps /\ Z e. X /\ A e. X ) -> ( Z H A ) e. X )
21	19, 12, 13, 20	syl3anc 1326	. . . 4 |- ( ( R e. RingOps /\ A e. X ) -> ( Z H A ) e. X )
22	3, 4	grporid 27371	. . . . 5 |- ( ( G e. GrpOp /\ ( Z H A ) e. X ) -> ( ( Z H A ) G Z ) = ( Z H A ) )
23	22	eqcomd 2628	. . . 4 |- ( ( G e. GrpOp /\ ( Z H A ) e. X ) -> ( Z H A ) = ( ( Z H A ) G Z ) )
24	18, 21, 23	syl2anc 693	. . 3 |- ( ( R e. RingOps /\ A e. X ) -> ( Z H A ) = ( ( Z H A ) G Z ) )
25	10, 17, 24	3eqtr3d 2664	. 2 |- ( ( R e. RingOps /\ A e. X ) -> ( ( Z H A ) G ( Z H A ) ) = ( ( Z H A ) G Z ) )
26	3	grpolcan 27384	. . 3 |- ( ( G e. GrpOp /\ ( ( Z H A ) e. X /\ Z e. X /\ ( Z H A ) e. X ) ) -> ( ( ( Z H A ) G ( Z H A ) ) = ( ( Z H A ) G Z ) <-> ( Z H A ) = Z ) )
27	18, 21, 12, 21, 26	syl13anc 1328	. 2 |- ( ( R e. RingOps /\ A e. X ) -> ( ( ( Z H A ) G ( Z H A ) ) = ( ( Z H A ) G Z ) <-> ( Z H A ) = Z ) )
28	25, 27	mpbid 222	1 |- ( ( R e. RingOps /\ A e. X ) -> ( Z H A ) = Z )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ran crn 5115   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   GrpOpcgr 27343  GIdcgi 27344   RingOpscrngo 33693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-1st 7168  df-2nd 7169  df-grpo 27347  df-gid 27348  df-ginv 27349  df-ablo 27399  df-rngo 33694

This theorem is referenced by:  rngonegmn1l  33740  isdrngo3  33758  0idl  33824  keridl  33831  prnc  33866